Geometry & Topology

Global Weyl groups and a new theory of multiplicative quiver varieties

Philip Boalch

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In previous work a relation between a large class of Kac–Moody algebras and meromorphic connections on global curves was established; notably the Weyl group gives isomorphisms between different moduli spaces of connections, and the root system is also seen to play a role. This involved a modular interpretation of many Nakajima quiver varieties, as moduli spaces of connections, whenever the underlying graph was a complete k–partite graph (or more generally a supernova graph). However in the isomonodromy story, or wild nonabelian Hodge theory, slightly larger moduli spaces of connections are considered. This raises the question of whether the full moduli spaces admit Weyl group isomorphisms, rather than just the open parts isomorphic to quiver varieties. This question will be solved here, by developing a multiplicative version of the previous approach. This amounts to constructing many algebraic symplectic isomorphisms between wild character varieties. This approach also enables us to state a conjecture for certain irregular Deligne–Simpson problems and introduce some noncommutative algebras (fission algebras) generalising the deformed multiplicative preprojective algebras (some special cases of which contain the generalised double affine Hecke algebras).

Article information

Geom. Topol., Volume 19, Number 6 (2015), 3467-3536.

Received: 19 August 2014
Accepted: 10 February 2015
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14L24: Geometric invariant theory [See also 13A50] 34M40: Stokes phenomena and connection problems (linear and nonlinear) 53D05: Symplectic manifolds, general 53D20: Momentum maps; symplectic reduction 53D30: Symplectic structures of moduli spaces

quiver variety complex symplectic quotient GIT quasi-Hamiltonian wild character variety Weyl group wild Riemann surface irregular curve Stokes local system


Boalch, Philip. Global Weyl groups and a new theory of multiplicative quiver varieties. Geom. Topol. 19 (2015), no. 6, 3467--3536. doi:10.2140/gt.2015.19.3467.

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